Moment map in Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks $2$

Question

In the paper Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks 2, in the part of the preliminaries the author considers a Hamiltonian action of the isometry group $G$ of $\mathbb{C}\mathbb{P}^n$ on $\mathbb{C}\mathbb{P}^n$. The action is described in terms of the dual of the lie algebra and then he claims that for $\xi$ in the lie algebra of a maximal torus $f_{\xi}(x):=\langle \Phi(x),\xi\rangle = \frac{x^T\xi x}{2\pi i\|x\|^2}$.Then it's claimed that we can pick a $\xi$ such that the flow $\psi_t$ of the vector field generated by it is periodic with period one , and that $\psi_t(\mathbb{R}\mathbb{P}^n)\cap \mathbb{R}\mathbb{P}^n = \operatorname{Crit}(f_{\xi})=n+1$.

Now I have tried to see this why this is true but I couldn't. I am only used to working with hamiltonian actions of torus $\mathbb{T}^n$ , where there isn't any talk about lie algebras and their dual, so I was wondering if we could describe the action of $G$ as an action of a torus so that I could check the statements.

Or if anyone could enlighten me why this statements are true I would appreciated it.

Thanks in advance.

Answer 1

The choice of $\xi$ amounts to choosing a Hamiltonian circle action on $\mathbb{CP}^n$ by isometries. I.e. choosing a suitable 1-parameter subgroup of the $PU(n+1,\mathbb{C})$.

Consider the Hamiltonian circle action $$z.[z_{0} : z_{1} : \ldots : z_{n}] = [z_{0} : z z_{1} : \ldots :z^{n} z_{n}] , $$

Which will have Hamiltonian $H=f_{\xi} = \frac{\sum_{i=0}^{n} i |z_{i}|^2}{\sum_{i=0}^{n} |z_i|^2}$. In the lie algebra of the maximal torus this is the point $\xi = (0,1,2,\ldots,n)$.

Then (as for any Hamiltonian circle action) the critical points are exactly the fixed points of the circle action; namely the $n+1$ points $[1:0:\ldots:0]$, $\ldots$, $[0:\ldots :1]$.

For the final claim suppose that $$ [z_{0} : z z_{1} : \ldots :z^{n} z_{n}] \in \mathbb{RP}^n \;\;\forall z .$$

Clearly the $n+1$ fixed points above satisfy this condition, we show there are no additional ones. Suppose that more than one of the homogenous co-ordinates is $0$, say $z_i$,$z_j$ with $i \neq j$. Taking $z=0$ which shows that we can start with all the $z_i$ real, if the two co-ordinates $z_i$, $z_{j}$ are non-zero and real with $i \neq j$; clearly $$\frac{z^i . z_{i}}{z^j . z_j}$$ can't be real for all $z \in S^1$ showing that the orbit of this point is not contained in $\mathbb{RP}^n$.